We present a theoretical analysis of Gaussian-binary restricted Boltzmann machines (GRBMs) from the perspective of density models. The key aspect of this analysis is to show that GRBMs can be formulated as a constrained mixture of Gaussians, which gives a much better insight into the model's capabilities and limitations. We show that GRBMs are capable of learning meaningful features both in a two-dimensional blind source separation task and in modeling natural images. Further, we show that reported difficulties in training GRBMs are due to the failure of the training algorithm rather than the model itself. Based on our analysis we are able to propose several training recipes, which allowed successful and fast training in our experiments. Finally, we discuss the relationship of GRBMs to several modifications that have been proposed to improve the model.
We extend the Chow-Liu algorithm for general random variables while the previous versions only considered finite cases. In particular, this paper applies the generalization to Suzuki's learning algorithm that generates from data forests rather than trees based on the minimum description length by balancing the fitness of the data to the forest and the simplicity of the forest. As a result, we successfully obtain an algorithm when both of the Gaussian and finite random variables are present.
We present a bound on the generalisation error of linear classifiers in terms of a refined margin quantity on the training set. The result is obtained in a PAC-Bayesian framework and is based on geometrical arguments in the space of linear classifiers. The new bound constitutes an exponential improvement of the so far tightest margin bound by Shawe-Taylor et al.  and scales logarithmically in the inverse margin. Even in the case of less training examples than input dimensions sufficiently large margins lead to nontrivial bound values and - for maximum margins - to a vanishing complexity term.Furthermore, the classical margin is too coarse a measure for the essential quantity that controls the generalisation error: the volume ratio between the whole hypothesis space and the subset of consistent hypotheses. The practical relevance of the result lies in the fact that the well-known support vector machine is optimal w.r.t. the new bound only if the feature vectors are all of the same length. As a consequence we recommend to use SVMs on normalised feature vectors only - a recommendation that is well supported by our numerical experiments on two benchmark data sets. 1 Introduction Linear classifiers are exceedingly popular in the machine learning community due to their straightforward applicability and high flexibility which has recently been boosted by the so-called kernel methods . A natural and popular framework for the theoretical analysis of classifiers is the PAC (probably approximately correct) framework which is closely related to Vapnik's work on the generalisation error . For binary classifiers it turned out that the growth function is an appropriate measureof "complexity" and can tightly be upper bounded by the VC (Vapnik-Chervonenkis) dimension .
The quality of training data is one of the crucial problems when a learning-centered approach is employed. This paper proposes a new method to investigate the quality of a large corpus designed for the recognizing textual entailment (RTE) task. The proposed method, which is inspired by a statistical hypothesis test, consists of two phases: the first phase is to introduce the predictability of textual entailment labels as a null hypothesis which is extremely unacceptable if a target corpus has no hidden bias, and the second phase is to test the null hypothesis using a Naive Bayes model. The experimental result of the Stanford Natural Language Inference (SNLI) corpus does not reject the null hypothesis. Therefore, it indicates that the SNLI corpus has a hidden bias which allows prediction of textual entailment labels from hypothesis sentences even if no context information is given by a premise sentence. This paper also presents the performance impact of NN models for RTE caused by this hidden bias.